Saturday, 29 January 2011

A proof of solipsism

I appreciate the comedy, of course, but I was always troubled by the maths of this Douglas Adamsian proof (from Restaurant):

It is known that there are an infinite number of worlds, simply because there is an infinite amount of space for them to be in. However, not every one of them is inhabited. Therefore, there must be a finite number of inhabited worlds. Any finite number divided by infinity is as near to nothing as makes no odds, so the average population of all the planets in the Universe can be said to be zero. From this it follows that the population of the whole Universe is also zero, and that any people you may meet from time to time are merely the products of a deranged imagination.

But that's not right, is it? I don't mean the thing about confusing 'average' figures with specific incidences; that doesn't bother me. I mean this bit: 'there are an infinite number of worlds, but not every one of them is inhabited. Therefore, there must be a finite number of inhabited worlds.' Subtract whatever number you like from infinity, you're still left with infinity. So the maths should go: 'there are an infinite number of worlds, and an infinite number of inhabited worlds (constituting an infinity of inhabitants). Infinity divided by infinity is ... one. So there is, on average, only one person in the cosmos.'

As a mathematician, I have to note that while Adams' calculation is wrong, so is yours. If you subtract an infinite number from an infinite number, you can get a finite result (you can get an infinite one as well).

Hi Alexey: thanks, yes. After posting this I thought to check whether infinity divided by itself really is '1', and I discovered that mathematicians seem to think it isn't. They're not sure what it is, mind you (according to this guy, it's either '2' or 'infinity'), but they're sure it's not one. Which only goes to show: infinity is a deeply counterintuitive matter.

That joke always bothered me as well - not just the math, though that especially For whatever strange juvenile reason, the whole people/exports/art/sex numbered run always seemed weirdly limp to me, except for the deft joke about the mirror...which of course relies on the run for its abrupt impact. It's all a bit too Adams-y for me, I think.

Maybe it's that he sets up this 'hey look, neat idea!' idea with a joke that's a leeeeeeeetle bit iffy (i.e. inf/N~=0), then carries on as if everything were OK?